Past exam of the chemistry course of the year 3 of the natural sciences course of the University of Cambridge / 2022 / Paper 2B / Section A / 32

The planar benzene structure has a sixfold principal axis normal to its plane, six transverse twofold axes, the molecular mirror plane and an inversion centre. Its molecular point group is $D_{6h}$, the D6h molecular point group.

The fulvene structure has an in-plane twofold axis through the exocyclic double bond and the opposite part of the five-membered ring. The molecular plane and a plane perpendicular to it containing that axis are both mirrors. There is no perpendicular sixfold axis or inversion centre. Its molecular point group is $C_{2v}$, the C2v molecular point group.

For hexa-1,5-diyne in the extended anti conformation drawn, inversion at the midpoint of the central carbon–carbon bond exchanges the two identical ends. A twofold axis through that midpoint perpendicular to the carbon skeleton plane, together with the mirror plane of that skeleton, completes the C2h molecular point group. The methylene hydrogens occur in the corresponding mirror-related pairs. Thus the answer is $C_{2h}$. The whole molecule is not linear simply because its terminal triple-bond fragments are linear; the conformation is essential to this assignment.

The planar three-radialene structure has a threefold axis normal to the plane, three in-plane twofold axes and the molecular plane as a horizontal mirror. Therefore its molecular point group is $D_{3h}$, the D3h molecular point group. Its threefold connectivity does not supply a sixfold axis.

The folded benzvalene cage has one twofold axis and two mirror planes containing it. The double-bonded edge distinguishes it from cages having higher rotational symmetry; it has neither an inversion centre nor a horizontal mirror perpendicular to the twofold axis. Its molecular point group is $C_{2v}$, the C2v molecular point group.